Embedded newform 804.2.u.b.43.17

• Label: 804.2.u.b.43.17
• Level: $804$
• Weight: $2$
• Character: 804.43
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $340$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(340\)
Relative dimension:	\(34\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			43.17
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.208996 + 1.39869i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-1.91264 + 0.584640i) q^{4} +(4.20210 + 0.604171i) q^{5} +(-0.932005 - 1.06366i) q^{6} +(0.989284 - 2.16623i) q^{7} +(-1.21746 - 2.55300i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{22}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.208996	+	1.39869i	0.147783	+	0.989020i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	4.20210	+	0.604171i	1.87924	+	0.270194i								0.984367	−	0.176131i	\(-0.0563582\pi\)
							0.894871	+	0.446325i	\(0.147267\pi\)
\(7\)	0.989284	−	2.16623i	0.373914	−	0.818758i	−0.625348	−	0.780346i	\(-0.715043\pi\)
							0.999262	−	0.0384120i	\(-0.0122299\pi\)
\(11\)	0.0869291	−	0.604605i	0.0262101	−	0.182295i	−0.972511	−	0.232858i	\(-0.925192\pi\)
							0.998721	+	0.0505625i	\(0.0161014\pi\)
\(13\)	−1.28868	+	4.38885i	−0.357416	+	1.21725i	0.563049	+	0.826423i	\(0.309628\pi\)
							−0.920465	+	0.390824i	\(0.872190\pi\)
\(17\)	3.85771	−	4.45204i	0.935633	−	1.07978i	−0.0610286	−	0.998136i	\(-0.519438\pi\)
							0.996662	−	0.0816422i	\(-0.0260165\pi\)
\(19\)	6.08413	−	2.77853i	1.39580	−	0.637439i	0.431463	−	0.902131i	\(-0.357998\pi\)
							0.964333	+	0.264692i	\(0.0852703\pi\)
\(23\)	1.46563	+	2.28056i	0.305604	+	0.475530i	0.959757	−	0.280833i	\(-0.0906108\pi\)
							−0.654152	+	0.756363i	\(0.726974\pi\)
\(29\)	−8.37493			−1.55519			−0.777593	−	0.628768i	\(-0.783559\pi\)
							−0.777593	+	0.628768i	\(0.783559\pi\)
\(31\)	1.12738	−	0.331028i	0.202483	−	0.0594544i	−0.178918	−	0.983864i	\(-0.557260\pi\)
							0.381401	+	0.924410i	\(0.375442\pi\)
\(37\)	−4.53459			−0.745481			−0.372741	−	0.927936i	\(-0.621582\pi\)
							−0.372741	+	0.927936i	\(0.621582\pi\)
\(41\)	−0.188008	−	0.162910i	−0.0293619	−	0.0254423i	0.640056	−	0.768328i	\(-0.278911\pi\)
							−0.669418	+	0.742886i	\(0.733457\pi\)
\(43\)	−0.416558	+	0.480734i	−0.0635245	+	0.0733112i	−0.786622	−	0.617434i	\(-0.788172\pi\)
							0.723098	+	0.690746i	\(0.242718\pi\)
\(47\)	4.61740	+	7.18482i	0.673517	+	1.04801i	0.994881	+	0.101057i	\(0.0322225\pi\)
							−0.321363	+	0.946956i	\(0.604141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.17	yes	340
4.3	odd	2		804.2.u.a.43.27	✓	340
67.53	odd	22		804.2.u.a.187.27	yes	340
268.187	even	22	inner	804.2.u.b.187.17	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.27	✓	340	4.3	odd	2
804.2.u.a.187.27	yes	340	67.53	odd	22
804.2.u.b.43.17	yes	340	1.1	even	1	trivial
804.2.u.b.187.17	yes	340	268.187	even	22	inner